File:PrimeNumberTheorem.png

Displayed is

• the number of primes less than or equal to x, labelled ${\displaystyle \Pi (x)}$ (red)
• an approximation using ${\displaystyle {\frac {x}{\ln {x}}}}$ (green)
• an approximation using the logarithmic integral ${\displaystyle \operatorname {Li} (x)}$ (blue)

Created with Maple: plot({[seq([x,pi(x)],x=0..100000)],x/ln(x),Li(x)-Li(2)},x=0..100000,Pi[x]=0..10000,color=[blue,green,red]);

Author: FredStober

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This work has been released into the public domain by its author, FredStober. This applies worldwide. In some countries this may not be legally possible; if so: FredStober grants anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law.